Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5×C8 · Schur multiplier (H2) = C22×C42 |
| direct product | C5×#7854 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C5 | C42 |
| commutator | C4 | C2×C4×C5×C8 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| socle | C22×C5 | C2×#95 |
| prime radical | C24×C5 | C22⋊C4 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C42×C5 ◁ G | |
714 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3,4,5)(6,7,9,12,10,13,8,11)(14,15)(16,22,24,28)(17,20)(18,21)(19,23,26,29)(25,27);(14,16)(15,19,21,26)(17,23)(18,24)(20,22,27,28)(25,29);(6,8,10,9)(7,11,13,12)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 842);